The cube condition which is at the heart of the developments in previous
sections is quite reminiscent of Gromov's condition for characterizing
non-positively curved cubical complexes [1]}. In order to
make the connection precise, one has to associate to every precubical set a
geodesic metric space (which is a cubical complex). Such a construction could be
performed abstractly as a geometric realization in the category of metric spaces
if this category were cocomplete... which is not the case. This leads us to
investigate a generalization of the notion of metric spaces (called
generalized metric spaces [2]} or Lawvere
metric spaces [3]} or hemi-metric
spaces [4]}) which form a cocomplete
category. Interestingly, these spaces also allow one to encode a notion of
“time direction” in the metric, as first noticed by
Grandis [5]}, [2]}, thus enabling one to
formulate a metric semantics for concurrent programs, which is not only able to
encode the direction of time but also the duration of the elapsed time during an
execution.
The use of non-positive curvature has been used by Ghrist in order to study
configuration spaces and cubical complexes [7]}, [8]}, [9]}.
